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Mathematics Learning Instrument: A Correlation and Means Difference
Comparison with Federal Accountability Algebra Test Scores
Jeffrey C. Lear
Whitehall-Coplay School District, PA, United States
Abstract
Mathematics learning outcomes have specific
cognitive development implications [2], [9]. Key
contributions in ‘structure sense’ studies build upon a
theoretical manuscript detailing a conceptualization
of ‘thinking mathematically’ using symbols (i.e.
‘symbol sense’) [1], [6], [12], [13]. Likewise, from a
theoretical basis, this manuscript reports and extends
an initial conference presentation of the instrument:
Algebra Concept Inventory to Measure Metric Sense
(ACIMMS) [14]. This paper includes the complete
instrument as well as developmental details,
particularly; (a) a definition of ‘metric sense’ [9], (b)
specific roles of mini-experts in Second Generation
Instructional Design (ID2) [11], and, (c) creates a
mathematics instrument using ‘metric sense’ to
inventory five subject specific domain knowledge base
items [5]. Selected applications of the instrument’s
constructs are illustrated in equations conventionally
found in trigonometry and statistics courses. Upper
classmen from a United States high school form a
sample for a correlation and means difference study
between federal accountability exam scores and
ACIMMS algebra instrument data. Confirmed are
findings from a bounded literature review [9], that
concurrent use of all five metric sense
conceptualizations are not extensively utilized by
secondary students in manipulating mathematical
expressions [6]. Despite time discrepancies of up to
three years between results of ACIMMS instrument
data and independent federal algebra accountability
exam scores, a significant positive correlation is
found. Correct answers on each instrument question,
which forms two respective groups of students’
independent algebra accountability exam scores,
reveal a clear need for students’ persistent
understanding of all five metric sense
conceptualizations. Recommendations are made for
teaching and further mathematics learning
investigations.
1. Introduction
This theoretical manuscript presents a definition
of metric sense using a metric space [12].
Mathematics learning requires a continual
examination of conceptual understanding while
executing each step of procedural approaches to
problem solving [2]. Recent research has examined
the effect of using brackets in procedural processes to
group structures according to an order of operations
[6]. Other studies compare written forms of
expressions to examine student comprehension of
structure (i.e. structure sense) [1], [10].
In this paper it is contended that procedural
processes are concurrent with structure
comprehension in making sense of mathematics [10],
[13]. Theoretically, the inherent mathematical
structure of a metric space will define the algebraic
structures mathematics students engage when using
procedures to rearrange groupings of symbols to solve
problems [12]. Five horizontal integrated algebra
knowledge base elements are hypothesized as
concurrently necessary to simplify expressions, solve
equations, and build skills in mathematics learning
[2], [5]. The distance function relating elements of a
metric space (e.g. variable representations among
symbols in algebra) is used in an analysis to justify
each integrated algebra knowledge base element.
Findings include an Algebra Concept Inventory to
Measure Metric Sense and two examples which
extend the research constructs to post algebra
structures (see Tables 3, 4, 2, and Figures 1a, 1b
respectively). The instrument is consistent with
current lines of inquiry in mathematics learning and
may compliment instrumentation developed in studies
in Newtonian physics [3], [4].
2. Metric Space and Metric Sense
The most familiar metric space used to develop
elementary algebraic skills is the Euclidean Space [8].
The Euclidean plane (i.e. Cartesian Plane RxR) is
used as a representation for learning algebraic
International Journal for Cross-Disciplinary Subjects in Education (IJCDSE), Volume 10, Issue 3, September 2019
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concepts where two real number lines intersect at an
origin. The representation provides access to
instruction on two variables x and y. The following
definition of a metric space theoretically supports the
development of instrument design and element
horizontal integration into a specific mathematics
domain knowledge base [5], [12].
A metric space is a set R with a distance function
(the metric d) that, for every two points x,y in R, gives
the distance between them as a nonnegative real
number d(x,y). A metric space must also satisfy
1. d(x,y) = 0 if and only if x = y
2. d(x,y) = d(y,x) and
3. the triangle inequality d(x,y) + d(y,z) >=
d(x,z).
“A Euclidian Space may be viewed as a vector space
with the usual metric (distance) explored in secondary
mathematics education. This focus on mathematical
conceptual understanding, a refinement of structure
sense within mathematical thinking, herein termed
metric sense; is hypothesized to be instrumental in
building empirical knowledge in” … mathematics
instruction … “and the assessment of learning
outcomes” [8], [9].
3. Metric sense as horizontal integrated
algebra knowledge base elements
A purpose of this paper is to specify algebra
knowledge base elements impacting future
knowledge production in mathematics learning.
Measuring metric sense cognitive skills is inherently
measuring proficiency in identifying and correctly
interpreting metric space elements [9], [12]. The
following delineates five measurable constructs of
metric sense conceptualization (MS1-MS5)
throughout each step in an equation and its procedural
solution (see Table 1).
1. MS1 - identify all the unary, binary, or other
operations
2. MS2 – identify the main operation (MO) of
an expression within an equation; utilizing
all the expression, not missing anything
3. MS3 - correctly execute the inverse
operation resulting in an equivalent equation
4. MS4 – identify all positive and negative
symbols in an expression as either only part
of a coefficient of a term, or concurrently as
a coefficient symbol and a binary operation
(i.e. sum and difference)
Table 1. Algebraic example identifying all metric sense elements in each solution step
+𝑥2 − 4
+10= +6
MS3 {apply inverse,
multiply}
MS2 MO quotient
MS5
Quotient of (difference and 10)
(Square and 4)
x
MS1 - Two binary operations (-, /)
and one unary operation (square)
MS4 – Three positive symbols as
coefficients on terms; one negative
symbol as symbol on 4 having dual
purpose as binary operation
difference/subtraction (total 4)
+𝑥2 − 4 = +60
MS3 {apply inverse add}
MS2 MO difference
MS5
Difference (Square and 4)
x
MS1 - One binary operation (-) and
one unary operation (square)
MS4 – Two positive symbols as
coefficients on terms; one negative
symbol as symbol on 4 having dual
purpose as binary operation
difference/subtraction (total 3)
+𝑥2 = +64
MS3 {apply inverse√𝑥2 =
√64}
MS2 MO square
MS5
(Square)
x
MS1 - one unary operation
MS4 – Two positive symbols as
coefficients on terms; no positive or
negative symbols having dual
purpose as binary operations (total 2)
+𝑥 = +8 * MS4 – Two positive symbols as
coefficients on terms; no positive or
negative symbols having dual
purpose as binary operations (total 2)
Note. * Positive result for illustration
International Journal for Cross-Disciplinary Subjects in Education (IJCDSE), Volume 10, Issue 3, September 2019
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5. MS5 – identify in writing, the hierarchy order of
operations beginning with the main operation
(MO), throughout the entire expression, detailing
all operands of unary and binary operations.
4. A relationship between metric sense
knowledge elements and a mathematical
metric space
The hypothesis of this manuscript is: a student
who has a high level of metric sense in the five
horizontal integrated algebra knowledge base
elements has a knowledge base necessary to simplify
expressions, solve equations, and build knowledge in
a metric space setting.
5. Theoretical analysis of the hypothesis
Represented here are theoretical justifications for
the hypothesis [12].
MS1, MS4: A metric space requires, for every
two points x,y in R, the triangle inequality d(x,y) +
d(y,z) >= d(x,z) holds. Therefore, the summation, and
its inverse binary operation, difference, must be
identifiable as independent or concurrent uses of the
+ and – symbols respective of an element’s distance
from either side of zero on a real number line.
MS2, MS5: Novotná and Hoch [12, p. 95] find
“An important feature of structure sense is the
substitution principle, which states that if a variable or
parameter is replaced by a compound term (product or
sum), or if a compound term is replaced by a
parameter, the structure remains the same.”
Therefore, a main operation (MO) of an expression
within an equation, utilizing all the expression, not
missing anything; incorporates possible compound
terms where in a metric space, for every two points x,
y in R, gives the distance between them, d(x, y).
MS3: The relationship between an operation’s
identity element and inverse elements of a metric
space are identifiable for every element in the
continuous set R [8].
An extended application of the five measurable
components of metric sense conceptualization, with
the same theoretical justifications, can be applied to
post algebra mathematics learning [12]. An
application of conceptualizing metric sense in a
trigonometric equation is provided (see Table 2).
Although introduced in algebra, it cannot be over
emphasized that continual conceptualization and
persistent application of the five horizontal integrated
algebra knowledge base elements is critical to
mathematics learning throughout secondary and
tertiary education.
Table 2. Trigonometric example identifying metric sense knowledge base elements
+sin(+𝜋 − 𝑥) = −2
Normally:
sin(𝜋 − 𝑥) = −2
MS3 {apply inverse sin-1}
MS2 MO sin function {unary}
MS5
MS1 - One binary operation (-) and
one unary operation, sin ( )
MS4 – Three positive/negative
symbols as coefficients on terms,
One negative symbol as a symbol on
x with dual purpose as a binary
operation difference (total 4)
Note. sin-1 is the inverse function of the sine function; first solution step provided for illustration
An additional application of metric sense and
structure sense is presented for student learning on the
proportional minimum sample size n formula from a
familiar margin of error (E) equation (see Figure 1a
and 1b). The classroom presentation episode is a
successful example of turning theory into practice.
6. A mathematics instrument using
‘metric sense’ to inventory algebra subject
specific domain knowledge base elements
Expressions typical in linear and quadratic
equations are analyzed in ten questions of an Algebra
Concept Inventory to Measure Metric Sense (see
Table 3 and 4). Level appropriate vocabulary is
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Table 3. Algebra Concept Inventory to Measure Metric Sense p. 1
Answer questions 1-5 using the equation below:
+2(+𝑥 − 4)
+3+ 5 = 0
Normally 2(𝑥 − 4)
3+ 5 = 0
1. Which is the main operation of the expression on the left side of the equation;
utilizing all of the left expression, not missing anything, considering the order of
operations A. + symbol before the x B. ÷ division bar
C. + symbol before the 5 D. - symbol before the 4
2. Which is correct about the expression on the left side of the equation A. four positive symbols on terms; one negative
symbol also as subtraction (total five)
B. one positive symbol on terms; one negative
symbol also as subtraction; three positive
symbols also as addition (total five)
C. two positive symbols on terms; one negative
symbol also as subtraction; two positive
symbols also as addition (total five)
D. three positive symbols on terms; one
negative symbol also as subtraction; one
positive symbol also as addition (total five)
3. Which is accurate about the expression on the left side of the equation,
considering the order of operations
A.
B.
C.
D.
4. Which is accurate about the expression on the left side of the equation A. It has one binary operation B. It has four binary operations
C. It has five binary operations D. It has no binary operations
5. What are the steps in solving the equation by reversing the main operation of the
expression on the left side of the equation; A. First subtract each side by 5 followed by
multiplying by 3 as the next step
B. First subtract each side by 5 followed by
adding by 4 as the next step
C. First subtract each side by 4 followed by
multiplying by 3 as the next step
D. First add each side by 5 followed by dividing
by 4 as the next step
Note. Coefficient symbols are shown and introductory vocabulary used for inventory instrument measurement.
Table 4. Algebra Concept Inventory to Measure Metric Sense p. 2
International Journal for Cross-Disciplinary Subjects in Education (IJCDSE), Volume 10, Issue 3, September 2019
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Answer questions 6-10 using the equation below:
−6 + √(−2)2 − 4(+3)(−𝑥) = 0 Normally
−6 + √(−2)2 − 4(3)(−𝑥) = 0
6. Which is the main operation of the expression on the left side of the equation;
utilizing all of the left expression, not missing anything, considering the order of
operations A. - symbol before the 4 B. + symbol after the 6
C. 2 symbol (square) D. √ symbol (square root)
7. Which is correct about the expression on the left side of the equation A. one negative symbol on terms; one positive
symbol on terms; two positive symbols also as
addition; two negative symbols also as
subtraction (total six)
B. two negative symbols on terms; one positive
symbol on terms; one positive symbol also as
addition; two negative symbols also as
subtraction (total six)
C. three negative symbols on terms; one positive
symbol on terms; one positive symbol also as
addition; one negative symbol also as
subtraction (total six)
D. one negative symbol on terms; one positive
symbol on terms; one positive symbol also as
addition; three negative symbols also as
subtraction (total six)
8. Which is accurate about the expression on the left side of the equation,
considering the order of operations
A.
B.
C. D.
9. What is accurate about the expression on the left side of the equation A. It has 6 binary and 1 unary operation B. It has 4 binary and 1 unary operations
C. It has 1 binary and 4 unary operations D. It has 4 binary and 2 unary operations
10. What are the steps in solving the equation by reversing the main operation of the
expression on the left side of the equation; A. First add each side by 6 followed by squaring
each side as the next step
B. First square each side followed by adding
each side by 6
C. First add each side by -2 followed by squaring
each side as the next step
D. First square each side followed by adding
each side by 2
Note. Coefficient symbols are shown and introductory vocabulary used for inventory instrument measurement.
International Journal for Cross-Disciplinary Subjects in Education (IJCDSE), Volume 10, Issue 3, September 2019
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Figure 1a and 1b. Metric sense instruction as constructs on confidence intervals for proportions
integrated in the instrument which is conventionally
familiar to beginning algebra students. Some latent
coefficient symbols are present to accentuate
structure sense knowledge base element
identification while using the instrument (i.e. MS4,
see Tables 1- 4). It is recommended that instructors
use the instrument customized and adapted to their
individual needs. Instructors wishing to use the
instrument to find a significance between data with
other assessments in algebra may need to determine
pre and post testing variables according to their
particular school setting. Additional delineations
may need to be specified when reporting on findings
using the instrument.
7. A sample for a correlation and means
difference calculation
Upper classmen from a United States high
school provide sample data for a correlation and
means difference study between their federal
accountability algebra exam scores and the
independent ACIMMS algebra instrument scores.
Convenience sampling is used, as each student
within the school has a school administrated federal
accountability algebra exam score. Administrative
permissions also warrant this sampling technique.
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8. Methodology
Approved by administration, students (n = 49)
voluntarily participated in the instrument data
collection, for credit, a few weeks before graduation
as a normal mathematics activity included in their
course work study. The majority of students
received a federal accountability exam score up to
three years prior, as a summative assessment at the
conclusion of an algebra 1 course. The researcher
determined that the time laps and participation in
other mathematics courses allows for instrument
measurement of retained algebraic metric sense
understanding (i.e. use of the five algebra base
elements MS1-MS5). Some vocabulary information
was determined essential and was provided
graphically as an additional page:
Unary Operations
√
⬚2
⬚3
Binary Operations
⬚ + ⬚
⬚ - ⬚
⬚ x ⬚
⬚ ÷ ⬚
ACIMMS algebra instrument scores were recorded
for each student, for each question. The instrument’s
range is 0-10, one point for each of the 10 multiple
choice question answered correctly. The State’s
federal accountability exam ranges for the following
four categories, Below Basic, Basic, Proficient, and
Advanced; are respectively, 1200-1438, 1439-1499,
1500-1545, 1546-1800. Each student’s exam value
was recorded. Student identification labels replaced
any identifying information collected during the
study, immediately after being paired. Paired data
are plotted using a spreadsheet and a Pearson
correlation coefficient critical value is used to
determine correlation significance (see Figure 2).
For each of the ten questions data of the ACIMMS
instrument, based on correctness, two groups of
accountability exam scores were created (see Table
5). Ten independent two samples mean t-test(s)
assuming equal variances were calculated.
9. Conclusions
There are specific cognitive skills required in
representing and explaining structure sense within
the mathematical construct of a Euclidian Space [8],
[9]. Students’ skills can be quantified for instruction
in manipulations of algebraic equations, and for
formative or summative assessment. ACIMMS
instrument scores and psychometrically valid
federal accountability exam scores are significantly
correlated, r = .2418, p < .05 (see Figure 2).
Figure 2. Paired data plot
Of the ten ACIMMS questions, only questions
4, 5, 9, and 10 had student responses greater than
50% correct (see Table 5). These responses are
representative of MS1 and MS3 measurable
constructs of metric sense conceptualization for
linear and quadratic expressions. These are more
identification tasked MS1 and procedural MS3 in
students’ interactions with expressions. The
questions with student responses less than 50%
correct are representative of metric sense constructs
MS2, MS4, and MS5. These require a more
comprehensive understanding of entire expressions,
including symbol location, orders of multiple
operations, and symbol use comprehension (i.e.
symbol sense and metric sense).
Table 5. Accountability exam scores
Correct Incorrect ACIMMS n M (SD) n M (SD)
1 8 1508.50(36.23) 41 1520.02(43.32)
2 18 1524.11(46.06) 31 1514.68(40.03)
3 12 1531.25(46.63) 37 1513.89(40.32)
4* 25 1530.04(46.59) 24 1505.75(33.51)
5 36 1523.31(44.32) 13 1503.85(32.64)
6 10 1512.80(37.37) 39 1519.51(43.60)
7 21 1517.38(42.27) 28 1518.71(42.77)
8 18 1512.22(38.69) 31 1521.58(44.24)
9 28 1522.21(34.05) 21 1512.71(51.37)
10 33 1517.27(40.67) 16 1519.94(46.29)
Note. * significance t(1) = 2.09, p < .05.
Ten, two-samples mean t-test(s) assuming equal
variances determined that for only half of the
ACIMMS questions, the group of accountability
exam scores corresponding to answering the
question correctly, had a mean greater than the
group mean representing the question incorrectly
International Journal for Cross-Disciplinary Subjects in Education (IJCDSE), Volume 10, Issue 3, September 2019
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answered. It is notable that four of those correspond
to linear expressions questions rather than quadratic
expressions questions. The low correct to incorrect
ratio for all questions responses may explain this
accountability exam score result (see Table 5). It is
also notable that the question with the highest
correct : incorrect ratio is question 5, 36:13. This is
representative of students arriving at a procedural
first step response MS3, but far from a success rate
potential if they concurrently, persistently utilized
all five instrument conceptualizations successfully.
ACIMMS questions 4, 5, and 9 are identified as
corresponding to MS1 and MS3 measurable
constructs of metric sense conceptualization for
linear and quadratic expressions. Examination
indicates that these are more identification tasked
MS1 and procedural MS3 in students’ interactions
with expressions. For ACIMMS question number 4
(i.e. MS1 - identify all the unary, binary, or other
operations), the mean for the group of accountability
exam scores where the question was answered
correctly (M = 1530.04, SD = 46.59) reported
significantly higher scores than the other group (M
= 1505.75, SD = 33.51), t(1) = 2.09, p < .05 . This
significance is found, and may possibly be
explained, where student focus is on how many
binary (+,_-) separations are between elements of
binary operations, necessary in part, to begin with
the first procedural step (i.e. question number 5) in
manipulating a linear expression. The t-test results,
like the direct ratio results, suggest that more
comprehensive understanding and persistent use of
symbol sense and metric sense is required when
students interact with expressions.
10. Implications for further research
This manuscript provides the Algebra Concept
Inventory to Measure Metric Sense (ACIMMS
quantitative instrument) to measure mathematics
conceptual understanding using five measurable
constructs of a metric space. This inventory
instrument was inspired by reviewing research
manuscripts using the Force Concept Inventory
(FCI) instrument [9]. The FCI test use in empirical
studies has initiated substantial contributions to the
line of inquiry in physics education [3], [4].
Instrumentation measuring the five constructs of
metric sense conceptualization (MS1-MS5) has the
potential to contribute similarly to the line of inquiry
in mathematics education.
It is recommended that similar replication
correlation studies verify the relationship between
the variables of ACIMMS results and results
commonly obtained through United States federal
accountability exams. Replicating this study’s two-
samples independent means tests, under various
initial field conditions, may reveal consistent
conclusions that students emphasize procedures
over metric sense comprehension when interacting
with mathematical expressions. It is also
recommended that mixed methods be implored
along with the instrument where qualitative
approaches can shed more light on students’ use of
ACIMMS metric sense constructs.
Included in mathematics learning mixed
method studies, the instrument may provide
evidence of relationships across data sources or
research design approaches. It is also recommended
the instrument be used with the FCI instrument to
measure scientific reasoning ability; adding some
clarity to unexplained negative correlation findings
between specific physics learner sample subgroups
[3].
Finally, it is recommended, and cannot be
understated in the authors opinion, that instruction
toward comprehension of MS1 through MS5 be
concurrent and persistent in a comprehensive way
until students can correctly generate tree
illustrations as an expected outcome with extremely
high success rates (see question numbers 3 and 8 and
Figure 1b). It may be necessary to continue this
practice throughout secondary and tertiary education
for some mathematics students to routinely inspect
expressions by thinking mathematically [13].
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